Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 26.5 → 0.7

Time: 13.1s

Precision: binary64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

↓

\[\begin{array}{l} t_0 := \left(\frac{3655.1204654076414}{x} + \frac{y}{x \cdot x}\right) + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \frac{-130977.50649958357}{x \cdot x}\right)\\ t_1 := {x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 \cdot \frac{{x}^{4}}{t_1} + \left(\frac{x \cdot y}{t_1} + \left(\frac{z}{t_1} + \left(137.519416416 \cdot \frac{{x}^{2}}{t_1} + 78.6994924154 \cdot \frac{{x}^{3}}{t_1}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- x 2.0)
   (+
    (*
     (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y)
     x)
    z))
  (+
   (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x)
   47.066876606)))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (/ 3655.1204654076414 x) (/ y (* x x)))
          (+
           (fma x 4.16438922228 -110.1139242984811)
           (/ -130977.50649958357 (* x x)))))
        (t_1
         (+
          (pow x 4.0)
          (+
           47.066876606
           (+
            (* 263.505074721 (pow x 2.0))
            (+ (* 43.3400022514 (pow x 3.0)) (* x 313.399215894)))))))
   (if (<= x -3.9e+71)
     t_0
     (if (<= x 3.6e+52)
       (*
        (+ x -2.0)
        (+
         (* 4.16438922228 (/ (pow x 4.0) t_1))
         (+
          (/ (* x y) t_1)
          (+
           (/ z t_1)
           (+
            (* 137.519416416 (/ (pow x 2.0) t_1))
            (* 78.6994924154 (/ (pow x 3.0) t_1)))))))
       t_0))))

C

double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

↓

double code(double x, double y, double z) {
	double t_0 = ((3655.1204654076414 / x) + (y / (x * x))) + (fma(x, 4.16438922228, -110.1139242984811) + (-130977.50649958357 / (x * x)));
	double t_1 = pow(x, 4.0) + (47.066876606 + ((263.505074721 * pow(x, 2.0)) + ((43.3400022514 * pow(x, 3.0)) + (x * 313.399215894))));
	double tmp;
	if (x <= -3.9e+71) {
		tmp = t_0;
	} else if (x <= 3.6e+52) {
		tmp = (x + -2.0) * ((4.16438922228 * (pow(x, 4.0) / t_1)) + (((x * y) / t_1) + ((z / t_1) + ((137.519416416 * (pow(x, 2.0) / t_1)) + (78.6994924154 * (pow(x, 3.0) / t_1))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(Float64(3655.1204654076414 / x) + Float64(y / Float64(x * x))) + Float64(fma(x, 4.16438922228, -110.1139242984811) + Float64(-130977.50649958357 / Float64(x * x))))
	t_1 = Float64((x ^ 4.0) + Float64(47.066876606 + Float64(Float64(263.505074721 * (x ^ 2.0)) + Float64(Float64(43.3400022514 * (x ^ 3.0)) + Float64(x * 313.399215894)))))
	tmp = 0.0
	if (x <= -3.9e+71)
		tmp = t_0;
	elseif (x <= 3.6e+52)
		tmp = Float64(Float64(x + -2.0) * Float64(Float64(4.16438922228 * Float64((x ^ 4.0) / t_1)) + Float64(Float64(Float64(x * y) / t_1) + Float64(Float64(z / t_1) + Float64(Float64(137.519416416 * Float64((x ^ 2.0) / t_1)) + Float64(78.6994924154 * Float64((x ^ 3.0) / t_1)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision] + N[(-130977.50649958357 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 4.0], $MachinePrecision] + N[(47.066876606 + N[(N[(263.505074721 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(43.3400022514 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 313.399215894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e+71], t$95$0, If[LessEqual[x, 3.6e+52], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(4.16438922228 * N[(N[Power[x, 4.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(z / t$95$1), $MachinePrecision] + N[(N[(137.519416416 * N[(N[Power[x, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(78.6994924154 * N[(N[Power[x, 3.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.5
Target	0.7
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000001e71 or 3.6e52 < x

   1. Initial program 63.2

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   2. Simplified 60.8

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

   3. Taylor expanded in x around inf 0.8

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(3655.1204654076414 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right)\right) - \left(110.1139242984811 + 130977.50649958357 \cdot \frac{1}{{x}^{2}}\right)} \]

   4. Simplified 0.8

      \[\leadsto \color{blue}{\left(\frac{3655.1204654076414}{x} + \frac{y}{x \cdot x}\right) + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) - \frac{130977.50649958357}{x \cdot x}\right)} \]

   if -3.9000000000000001e71 < x < 3.6e52

   1. Initial program 2.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   2. Simplified 0.7

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

   3. Taylor expanded in y around 0 0.7

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(4.16438922228 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(137.519416416 \cdot \frac{{x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + 78.6994924154 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+71}:\\ \;\;\;\;\left(\frac{3655.1204654076414}{x} + \frac{y}{x \cdot x}\right) + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \frac{-130977.50649958357}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+52}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)} + \left(\frac{x \cdot y}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)} + \left(\frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)} + \left(137.519416416 \cdot \frac{{x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)} + 78.6994924154 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + x \cdot 313.399215894\right)\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{3655.1204654076414}{x} + \frac{y}{x \cdot x}\right) + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) + \frac{-130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))